Characteristic polynomial/Coprime factorization/Direct sum/Fact/Proof

Proof

Due to the Lemma of Bezout, there exist polynomials such that

Set and . Let . Due to the Theorem of Cayley-Hamilton , we have

Therefore, the image of belongs to the kernel of and vice versa. From

we can read off that the left-hand summand belongs to and the right-hand summand belongs to . Therefore, we have a sum decomposition, which is direct, since implies . For the -invariance of these spaces, see exercise. For , we have

that is, we have . Therefore, the restriction of to the kernel of is surjective, thus bijective.